General Capelli-type identities
Naihuan Jing, Yinlong Liu, Jian Zhang

TL;DR
This paper introduces a new method to derive a broad family of Capelli-type identities, generalizing classical and higher identities, and confirming several conjectures in the theory of noncommutative determinants.
Contribution
A novel approach to derive a comprehensive family of Capelli identities, unifying and extending previous results and conjectures in noncommutative algebra.
Findings
Derived generalized Capelli identities including classical and higher cases
Established generalized Turnbull's identities for symmetric and antisymmetric matrices
Confirmed conjectures of Caracciolo, Sokal, and Sportiello on Capelli identities
Abstract
The classical Capelli identity is an important determinantal identity of a matrix with noncommutative entries that determines the center of the enveloping algebra of the general linear Lie algebra, and was used by Weyl as a main tool to study irreducible representations in his famous book on classical groups. In 1996 Okounkov found higher Capelli identities involving immanants of the generating matrix of which correspond to arbitrary orthogonal idempotent of the symmetric group. It turns out that Williamson also discovered a general Capelli identity of immanants for in 1981. In this paper, we use a new method to derive a family of even more general Capelli identities that include the aforementioned Capelli identities as special cases as well as many other Capelli-type identities as corollaries. In particular, we obtain generalized Turnbull's identities for both…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
Topicsgraph theory and CDMA systems · Matrix Theory and Algorithms · Graph theory and applications
